Quantum trajectories in atom-surface scattering with single adsorbates: the role of quantum vortices

In this work, a full quantum study of the scattering of He atoms off single CO molecules, adsorbed onto the Pt(111) surface, is presented within the formalism of quantum trajectories provided by Bohmian mechanics. By means of this theory, it is shown that the underlying dynamics is strongly dominated by the existence of a transient vortitial trapping with measurable effects on the whole diffraction pattern. This kind of trapping emphasizes the key role played by quantum vortices in this scattering. Moreover, an analysis of the surface rainbow effect caused by the local corrugation that the CO molecule induces on the surface, and its manifestation in the corresponding intensity pattern, is also presented and discussed.     

Quantum trajectories for resonant scattering

Time‐dependent wave packet resonant scattering for the double square barrier has been studied in terms of Bohm quantum trajectories. The high transmission probability for the wave packet with a resonant energy can be explained by the behavior of the quantum trajectories under the influence of the relatively slow formation of a node within the first barrier. This node splits the trajectories into reflected and transmitted components. During this stage, many particle trajectories pass through the double‐barrier region and contribute to the transmitted part of the wave packet. Due to the transient nature of the nodes, trajectories in the reflected wave packet bunch together between the nodes for a finite period of time so that temporary structure (localization of particles and accompanying increase in the probability density) develops on small length scales. These calculations also show that the particles gain high momentum near the nodal points, and they reach a uniform momentum distribution after transmitting the barrier region. We have found that the presence of a node between the two barriers influences the different lifetimes of the quasi‐bound states. © 2001 John Wiley & Sons, Inc. Int J Quant Chem 81: 206–213, 2001     

Quantum trajectories in complex space: one-dimensional stationary scattering problems

One-dimensional time-independent scattering problems are investigated in the framework of the quantum Hamilton-Jacobi formalism. The equation for the local approximate quantum trajectories near the stagnation point of the quantum momentum function is derived, and the first derivative of the quantum momentum function is related to the local structure of quantum trajectories. Exact complex quantum trajectories are determined for two examples by numerically integrating the equations of motion. For the soft potential step, some particles penetrate into the nonclassical region, and then turn back to the reflection region. For the barrier scattering problem, quantum trajectories may spiral into the attractors or from the repellers in the barrier region. Although the classical potentials extended to complex space show different pole structures for each problem, the quantum potentials present the same second-order pole structure in the reflection region. This paper not only analyzes complex quantum trajectories and the total potentials for these examples but also demonstrates general properties and similar structures of the complex quantum trajectories and the quantum potentials for one-dimensional time-independent scattering problems.     

Resonance charge transfer in atom-surface scattering

We present a review of theoretical work on resonance charge transfer in atom-surface scattering. We consider resonance coupling of the atomic level to a wide conduction band, and to low-lying core states. In each case the models, and the methods available for the calculation of the probability of electron transfer are summarised and their applicability demonstrated by selected applications to scattering and sputtering experiments.     

Electron-hole pair excitation in atom-surface scattering

The inelastic scattering of a light atom (helium) from a metallic surface with creation or annihilation of electron-hole pairs is studied. Diagonal and off-diagonal matrix elements of the atom-metal interaction are discussed. The nonadiabatic contributions to the scattering are analyzed in terms of a statistical potential. Explicit calculations are performed for a jellium model at electron densities corresponding to Al and Na. The electron-hole pair excitation probabilities turn out to be of the order of 10^?5–10^?3.     

Multiphonon energy transfer in atom-surface scattering

The probability density P(ω,q) of transferring the energy ω and the parallel momentum q to the surface is determined by the maximum entropy subject to constraints procedure. The two important constraints are identified as the recoil energy and the mean parallel energy transfer. Having determined P(ω,q) one can evaluate all observable quantities, e.g., the triple differential reflection probability d³σ/dE_tdΩ or the trapping probability. A three-parameter model for d³σ/dE_tdΩ is derived by assuming a parameterized form for the recoil energy. This model may be regarded as an extension of the hard-cube model, because it reduces to the latter if the third parameter, the speed of sound c, is set to infinity. The comparison of the predicted velocity and angular distributions with recent experiments of Hurst et al. is excellent considering the simplicity of the model.     

Applications of symmetry in coupled channel atom-surface scattering calculations

It has been shown that a partial decoupling of the coupled channel equations for atom-surface scattering can be obtained by the use of symmetry and this greatly reduces the numerical èffort required to solve the equations. The method is illustrated for the systems Ne/W(110), Ne/LiF(001) in the simplified framework of a recently-proposed sudden approximation.     

Adsorption effects on resonant charge transfer in atom-surface scattering

The probability of positive ionization of an atom impinging on a solid surface is calculated by numerical solution of the equations of motion of the time-dependent molecular orbitals of the system. The model is applied to the scattering of Na atoms, in the energy range 50–1000 eV, from a W(110) surface. The presence of Na adatoms is considered and found to hinder the ionization process. This reduction in the charge-transfer probability arises because of the lowering of the surface work function brought about by increasing adatom concentration.     

Quantum mechanical generalized phase-shift approach to atom-surface scattering: a Feshbach projection approach to dealing with closed channel effects

We have developed a new method for solving quantum dynamical scattering problems, using the time-independent Schr?dinger equation (TISE), based on a novel method to generalize a "one-way" quantum mechanical wave equation, impose correct boundary conditions, and eliminate exponentially growing closed channel solutions. The approach is readily parallelized to achieve approximate N(2) scaling, where N is the number of coupled equations. The full two-way nature of the TISE is included while propagating the wave function in the scattering variable and the full S-matrix is obtained. The new algorithm is based on a "Modified Cayley" operator splitting approach, generalizing earlier work where the method was applied to the time-dependent Schr?dinger equation. All scattering variable propagation approaches to solving the TISE involve solving a Helmholtz-type equation, and for more than one degree of freedom, these are notoriously ill-behaved, due to the unavoidable presence of exponentially growing contributions to the numerical solution. Traditionally, the method used to eliminate exponential growth has posed a major obstacle to the full parallelization of such propagation algorithms. We stabilize by using the Feshbach projection operator technique to remove all the nonphysical exponentially growing closed channels, while retaining all of the propagating open channel components, as well as exponentially decaying closed channel components.     

An endogenous multiple scattering method for the calculation of the elastic scattering cross sections of electron shape resonances in polyatomic molecu

A procedure is described for obtaining the scattering potential for elastic electron—molecule scattering within the one-electron overlapping sphere multiple scattering Xα method. The method has been used to calculate the total elastic electron scattering cross sections for the nitrogen, ethylene and 1,3,5-trifluorobenzene molecules, which compare well with experimental data.     

Complex trajectories sans isochrones: quantum barrier scattering with rectilinear constant velocity trajectories

One of the major obstacles in employing complex-valued trajectory methods for quantum barrier scattering calculations is the search for isochrones. In this study, complex-valued derivative propagation method trajectories in the arbitrary Lagrangian-Eulerian frame are employed to solve the complex Hamilton-Jacobi equation for quantum barrier scattering problems employing constant velocity trajectories moving along rectilinear paths whose initial points can be in the complex plane or even along the real axis. It is shown that this effectively removes the need for isochrones for barrier transmission problems. Model problems tested include the Eckart, Gaussian, and metastable quadratic+cubic potentials over a variety of wave packet energies. For comparison, the "exact" solution is computed from the time-dependent Schrodinger equation via pseudospectral methods.     

Quantum trajectories in complex phase space: multidimensional barrier transmission

The quantum Hamilton-Jacobi equation for the action function is approximately solved by propagating individual Lagrangian quantum trajectories in complex-valued phase space. Equations of motion for these trajectories are derived through use of the derivative propagation method (DPM), which leads to a hierarchy of coupled differential equations for the action function and its spatial derivatives along each trajectory. In this study, complex-valued classical trajectories (second order DPM), along which is transported quantum phase information, are used to study low energy barrier transmission for a model two-dimensional system involving either an Eckart or Gaussian barrier along the reaction coordinate coupled to a harmonic oscillator. The arrival time for trajectories to reach the transmitted (product) region is studied. Trajectories launched from an "equal arrival time surface," defined as an isochrone, all reach the real-valued subspace in the transmitted region at the same time. The Rutherford-type diffraction of trajectories around poles in the complex extended Eckart potential energy surface is described. For thin barriers, these poles are close to the real axis and present problems for computing the transmitted density. In contrast, for the Gaussian barrier or the thick Eckart barrier where the poles are further from the real axis, smooth transmitted densities are obtained. Results obtained using higher-order quantum trajectories (third order DPM) are described for both thick and thin barriers, and some issues that arise for thin barriers are examined.     

Jost function description of elastic and few-body resonances

We discuss how the analysis of the zeros of the functions introduced by Jost in 1946, acting individually or collectively, provides a comprehensive framework for describing resonances in single and multichannel collisions. In particular, we propose a generalization of the Wigner threshold law that copes with some deviations recently observed in opening reaction channels. We also pay special attention to the appearance of zeros of the s-wave Jost function in the fourth quadrant of the complex momentum plane, as analysed by Nussenzveig in 1959 but erroneously ruled out in following studies.     

Polymer characterization: Quasi-elastic and elastic light scattering

Dynamic or quasi-elastic light scattering (LS) from polymers in solution arises from concentration fluctuations. With the aid of modern photomultipliers these can be followed as a function of time. A proper evaluation allows to study the center of mass motion and the dynamics of individual chains. The relevance of simultaneous recording of static and dynamic LS is emphazised. Various aspects are discussed in three main sections. In the first part basic relationships are reviewed. The particle scattering factor P(8), structure factor S(q,c) and osmotic compressibility RT(∂c/∂T) occuring in static LS are defined. The time correlation functions (TCF) in dynamic LS are described. The TCF of the scattered electric field contains the time dependent structure factor S(q,t) and the static structure factor S(q). The initial part of the TCF (short delay times) can be approximated by a cumulant expansion; the first cumulant is related to the translational diffusion coefficient D. The concentration dependence of D contains a thermodynamic and a hydrodynamic contribution where the thermodynamic part is identical with the osmotic compressibility in static LS. The second part deals with the behaviour of various polymeric architectures in dilute solutions. Two new structure sensitive parameters, C and = Rg/Rh, are introduced. Chain stiffness and branching are extensively discussed. In the third part properties of different macromolecular architectures in semi-dilute solution are considered. The inverse osmotic compressibility = osmotic modulus and the concentration dependence of the translational diffusion coefficient are discussed in the light of re-normalization group and scaling theories.     

A refined He-LiF(001) potential from selective adsorption resonances measured with high-resolution helium spin-echo spectroscopy

The authors have developed a new experimental approach for measuring gas-surface selective adsorption resonances with much higher energy resolution and over a wider range of kinematic conditions than has previously been possible. The technique involves using a 3He spin-echo spectrometer as a Fourier transform helium atom scattering apparatus. The authors applied the technique to the He-LiF(001) system. They developed a new empirical potential for the He-LiF(001) system by analyzing and refining the best existing potentials in the light of the new data set. Following an initial free-particle model analysis, the authors used exact close coupling scattering calculations to compare the existing potentials with the new experimental data set. Systematic differences are observed between the two. The existing potentials are modified by simple transformations to give a refined potential that is consistent with and fully reproduces the experimental data. Their technique represents a new approach for developing very high precision empirical potentials in order to test first principles theory.     

Quantum manifestations of classical trajectories in molecular systems

Quantum chaos, understood as the effect of the underlying classical dynamics on the stationary quantum properties in classically chaotic systems, is examined in two molecular floppy systems. Realistic models of two degrees of freedom for HO₂ and HCN/HNC are considered. The structure of the classical phase space is studied using Poincaré surfaces of section and the dynamical characteristics of the corresponding wave functions analyzed also in phase space with the aid of Husimi functions. Some wave functions show strong localization along periodic orbits. © 2002 John Wiley & Sons, Inc. Int J Quantum Chem, 2001     

Plasmon resonances on metal tips: understanding tip-enhanced Raman scattering

Calculations of the electric-field enhancements in the vicinity of an illuminated silver tip, modeled using a Drude dielectric response, have been performed using the finite difference time domain method. Tip-induced field enhancements, of application in "apertureless" Raman scanning near-field optical microscopy (SNOM), result from the resonant excitation of plasmons on the metal tip. The sharpness of the plasmon resonance spectrum and the highly localized nature of these modes impose conditions to better exploit tip plasmons in tip-enhanced apertureless SNOM. The effect of tip-to-substrate separation and polarization on the resolution and enhancement are analyzed, with emphasis on the different field components parallel and perpendicular to the substrate.     

Quasi-bound resonances in electron scattering by small metal clusters

A quantal calculation to evaluate theoretically the elastic cross section of electrons scattered by metal clusters is presented. The method is applied to the elastic dispersion of low energy electrons (up to 5 eV) from the spherical sodium clusters Na₈ and Na₂₀. Strong resonances in the total cross sections are found for some energies which are closely related to the existence of quasi-bound states, i.e. states completely bound in the classical limit.     

Barrier scattering with complex-valued quantum trajectories: taxonomy and analysis of isochrones

To facilitate the search for isochrones when using complex-valued trajectory methods for quantum barrier scattering calculations, the structure and shape of isochrones in the complex plane were studied. Isochrone segments were categorized based on their distinguishing features, which are shared by each situation studied: High and low energy wave packets, scattering from both thick and thin Gaussian and Eckart barriers of varying height. The characteristic shape of the isochrone is a trifurcated system: Trajectories that transmit the barrier are launched from the lower branch (T), while the middle and upper branches form the segments for reflected trajectories (F and B). In addition, a model is presented for the curved section of the lower branch (from which transmitted trajectories are launched), and important features of the complex extension of the initial wave packet are identified.     

Quantum mechanical study on plasmon resonances in small ring clusters

The collectivity of the electronic motion in small sodium clusters with ring structure is studied by time‐dependent density functional theory. The formation and development of collective resonances in the absorption spectra were obtained as a function of the ring radius. In small ring clusters, besides the lower‐energy mode and the higher‐energy mode, there is another plasmon resonance mode, that is, the reverse two‐dipole mode. For the reverse two‐dipole mode, the formations of these two dipoles are due to the external field inducement and the shielding effect, although the resonant excitation is mainly due to the coupling effect of the electrons of these two dipoles. © 2012 Wiley Periodicals, Inc.